Hydrodynamic Limit for Exclusion Processes

Abstract

The exclusion process, sometimes called Kawasaki dynamics or lattice gas model, describes a system of particles moving on a discrete square lattice with an interaction governed by the exclusion rule under which at most one particle can occupy each site. We mostly discuss the symmetric and reversible case. The weakly asymmetric case recently attracts attention related to KPZ equation; cf. Bertini and Giacomin (Commun Math Phys 183:571–607, 1995) for a simple exclusion case and Goncalves and Jara (Arch Ration Mech Anal 212:597–644, 2014) for an exclusion process with speed change, see also Goncalves et al. (Ann Probab 43:286–338, 2015), Gubinelli and Perkowski (J Am Math Soc 31:427–471, 2018). In Sect. 1, as a warm-up, we consider a simple exclusion process and discuss its hydrodynamic limit and the corresponding fluctuation limit in a proper space–time scaling. From this model, one can derive a linear heat equation and a stochastic partial differential equation (SPDE) in the limit, respectively. Section 2 is devoted to the entropy method originally invented by Guo et al. (Commun Math Phys 118:31–59, 1988). We consider the exclusion process with speed change, in which the jump rate of a particle depends on the configuration nearby the particle. This gives a non-trivial interaction among particles. We study only the case that the jump rate satisfies the so-called gradient condition. The hydrodynamic limit, which leads to a nonlinear diffusion equation, follows from the local ergodicity or the local equilibrium of the system, and this is shown by establishing one-block and two-block estimates. We also discuss the fluctuation limit which follows by showing the so-called Boltzmann–Gibbs principle. Section 3 explains the relative entropy method originally due to Yau (Lett Math Phys 22:63–80, 1991). This is a variant of GPV method and gives another proof for the hydrodynamic limit. The difference between these two methods is as follows. Let $$N^d$$ be the volume of the domain on which the system is defined (typically, d-dimensional discrete box with side length N) and denote the (relative) entropy by H. Then, H relative to a global equilibrium behaves as $$H=O(N^d)$$ (or entropy per volume is O(1)) as $$N\rightarrow \infty .$$ GPV method rather relies on the fact that the entropy production I, which is the time derivative of H, behaves as $$O(N^{d-2})$$ so that I per volume is o(1), and this characterizes the limit measures. On the other hand, Yau’s method shows $$H=o(N^d)$$ for H relative to local equilibria so that the entropy per volume is o(1) and this proves the hydrodynamic limit. In Sect. 4, we consider Kawasaki dynamics perturbed by relatively large Glauber effect, which allows creation and annihilation of particles. This leads to the reaction–diffusion equation in the hydrodynamic limit. We discuss especially the equation with reaction term of bistable type and the problem related to the fast reaction limit or the sharp interface limit leading to the motion by mean curvature. We apply the estimate on the relative entropy due to Jara and Menezes (Non-equilibrium fluctuations of interacting particle systems, 2017; Symmetric exclusion as a random environment: invariance principle, 2018), which is actually obtained as a combination of GPV and Yau’s estimates. This makes possible to study the hydrodynamic limit for microscopic systems with another diverging factors apart from that caused by the space–time scaling.